MODERN COSMOLOGY

Quantifying large-scale structure 83

Figure 2.15.The correlation function of galaxies in the semi-analytical simulation of an
LCDM universe by Bensonet al(2000a).

proposed the alternative form

ρ/ρb=

c

y^3 /^2 ( 1 +y^3 /^2 )

(r<rvir); y≡r/rc.

Using this model, it is then possible to calculate the correlations of the nonlinear
density field, neglecting only the large-scale correlations in halo positions. The
power spectrum determined in this way is shown in figure 2.16, and turns out to
agree very well with the exact nonlinear result on small and intermediate scales.
The lesson here is that a good deal of the nonlinear correlations of the dark matter
field can be understood as a distribution of random clumps, provided these are
given the correct distribution of masses and mass-dependent density profiles.
How can we extend this model to understand how the clustering of galaxies
can differ from that of the mass? There are two distinct ways in which a degree
of bias is inevitable:

(1) Halo occupation numbers. For low-mass halos, the probability of obtaining

anL∗galaxy must fall to zero. For halos with mass above this lower limit,